Journal of Mathematical Imaging and Vision

, Volume 19, Issue 3, pp 237–253

The Classical Theory of Invariants and Object Recognition Using Algebraic Curve and Surfaces


  • Hakan Civi
    • AT Kearny Consulting
  • Colin Christopher
    • Department of MathematicsBogazici University Bebek
  • Aytul Ercil
    • Engineering and Natural SciencesSabanci University

DOI: 10.1023/A:1026233121583

Cite this article as:
Civi, H., Christopher, C. & Ercil, A. Journal of Mathematical Imaging and Vision (2003) 19: 237. doi:10.1023/A:1026233121583


Combining implicit polynomials and algebraic invariants for representing and recognizing complicated objects proves to be a powerful technique. In this paper, we explore the findings of the classical theory of invariants for the calculation of algebraic invariants of implicit curves and surfaces, a theory largely disregarded in the computer vision community by a shadow of skepticism. Here, the symbolic method of the classical theory is described, and its results are extended and implemented as an algorithm for computing algebraic invariants of projective, affine, and Euclidean transformations. A list of some affine invariants of 4th degree implicit polynomials generated by the proposed algorithm is presented along with the corresponding symbolic representations, and their use in recognizing objects represented by implicit polynomials is illustrated through experiments. An affine invariant fitting algorithm is also proposed and the performance is studied.

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© Kluwer Academic Publishers 2003